Simple signaling networks

Figure 1A shows a schematic of a simple signaling network, composed of two pathways, X and Y. Each pathway in the network has a unique input and a unique output. The input for pathway X is designated x₀, and can be taken to represent both the signal itself and its receptor; the input for pathway Y is designated y₀. We assume that the network is exposed to only one signal at a time. The activated species of the downstream elements of pathway X are designated x₁ and x₂, those of pathway Y are y₁ and y₂. When a signal x₀ is present, x₀ 'activates' (i.e. causes the production of) component x₁ (which might be a protein kinase or a kinase cascade); x₁ in turn activates x₂ (which might be a terminal kinase or a target transcription factor). Pathway components are also deactivated: proteins that are activated by being phosphorylated by a protein kinase are deactivated when that phosphate group is removed by a protein phosphatase, for instance. Figure 1B shows the typical mound-shaped curve of the time course of activation of a given component in response to a square-pulse input signal. The area under such a curve for the final component of a given pathway can be taken as a measure of that pathway's total output. Let us denote the total output of pathway X when the cell is exposed to an input signal x₀ as X_out|X_in (read as 'X output given X input', or simply 'X given X').

Figure 1

A simple network with crosstalk. (A) The network consists of two pathways, X and Y, that are interconnected because component y₁ activates target x₂. (B) Output in response to pulse of signal x₀ (top) or y₀ (bottom). (C) Depiction of the ratios equal to the specificity of pathway Y and the fidelity of pathway X.

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Interconnections between pathways often serve a useful purpose (Schwartz and Baron, 1999), but here we concern ourselves with undesirable crosstalk, or 'leaking'. In the Figure 1 network, pathway Y leaks into pathway X, because kinase y₁ is somewhat lacking in substrate selectivity: in addition to phosphorylating its correct target y₂, it also phosphorylates the incorrect target x₂. Hence, when the network is stimulated by signal y₀, in addition to the authentic output Y_out|Y_in there is some spurious output X_out|Y_in.

Definitions of specificity and fidelity

We define the specificity of cascade X as the ratio of its authentic output to its spurious output:

Thus (as in Figure 1), if pathway X is activated by a given signal and this does not affect the output from pathway Y, the specificity of X with respect to Y in response to that signal is infinite, or complete.

Similarly, the specificity of cascade Y is of the form

In Figure 1, since the action of signal y₀ will result in some output from X, the specificity of Y with respect to X is finite (see Figure 1C). Indeed, if S_Y were less than 1, it would mean that the signal for Y was actually promoting the output of pathway X more than its own output.

The fidelity of a pathway is its output when given an authentic signal divided by its output in response to a spurious signal (see Figure 1C):

Thus, a pathway that exhibits fidelity (i.e. F>1) is activated more by its authentic signal than by others. In contrast, if a pathway has fidelity of less than 1, it is activated more by another pathways' signal than it is by its own. In the Figure 1 network, F_Y is complete, while F_X is finite.

Cascades that share components

In many cases, two signaling pathways share one or more common elements (see Figure 2A). One well-known example is in mammalian PC12 cells, where treatment with epidermal growth factor (EFG) causes the cells to proliferate, whereas treatment with nerve growth factor (NFG) causes the cells to differentiate and sprout neurites, yet both growth factors signal through the same MAPK cascade (Marshall, 1995). Another example is in baker's and brewer's yeast (Saccharomyces cerevisiae), where three distinct signaling pathways (mating, invasive growth and osmotic stress response) share elements of the same MAP kinase cascade (van Drogen and Peter, 2002). Experimental data indicate that pathways can be well insulated from one another despite sharing components: treatment of PC12 cells with EGF does not cause them to sprout neurites, and stimulation of yeast with mating pheromone does not activate the stress response, for example (Schaeffer and Weber, 1999; van Drogen and Peter, 2002; Vaudry et al, 2002).

Figure 2

Signaling network with shared components. (A) The 'basic architecture'. Component x₁ is common to pathways X and Y. Although the desired route of signaling is for x₀ to activate x₂ and not y₂, and y₀ to activate y₂ and not x₂, this cannot be achieved with specificity and fidelity for this network. (B, C) Numerical simulations of signaling through this network under various sets of parameter values. Values that increase S_X reciprocally decrease S_Y, and values that increase F_X reciprocally decrease F_Y. Shown are the values of outputs x₂ and y₂ in response to inputs x₀ and y₀, applied separately as square pulses of magnitude 1 and duration 1. In panel B, both the specificity and fidelity of cascade X are larger than those of cascade Y. Parameter values are a₁=2, b₁=1, a₂=2, b₂=1, d₁=d₂^x=d₂^y=1. We have S_X=2, F_X=2, S_Y=0.5, F_Y=0.5. In panel C, the specificity of cascade X is higher than that of cascade Y whereas the opposite holds for fidelity values: F_X<F_Y. Parameter values are a₁=1, b₁=2, a₂=2, b₂=1, d₁=d₂^x=d₂^y=1. This yields S_X=2, F_X=0.5, S_Y=0.5, F_Y=2. (D, E) Two insulating mechanisms that can augment specificity: (D) compartmentalization; (E) the action of a scaffold protein.

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This class of networks can be represented by the 'basic architecture' shown in Figure 2A. Here, the parameters a₁ and a₂ are activation rate constants for pathway X; a₂ is the rate at which kinase x₁ activates (phosphorylates) target x₂. Similarly, b₁ and b₂ are activation rate constants for pathway Y. Finally, d₁^x, d₂^x and d₂^y are deactivation rate constants, and can be thought of as representing phosphatase activity, for example. Assuming that both pathways in the network are weakly activated (Heinrich et al, 2002; Chaves et al, 2004), the network can be modeled by a simple system of three linear ordinary differential equations (ODEs) that can be solved to yield precise analytical expressions for pathway outputs, specificities and fidelities in terms of the network parameters (see Table I; also, see Supplementary information for detailed solutions). It can be seen (see Table I) that the specificities of the pathways are independent of the signal strength, and indeed of all parameters that lie upstream or at the level of the shared component. Pathway fidelities, in contrast, depend strongly upon the relative signal strengths and upon the values of upstream parameters. Hence, the two performance metrics, specificity and fidelity, depend on different characteristics of network design. Indeed, it is easy to choose parameters that provide one pathway with specificity but not fidelity.

Table 1: Equations and solutions for the networks analyzed in this paper^a

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We define network specificity as the product of the pathway specificities:

(Note that network fidelity, the product of the pathway fidelities, is always exactly equal to network specificity.) S_network provides an indication of the specificity intrinsic in the network architecture. Intuitively, it would seem that the basic architecture does not possess intrinsic specificity. Indeed, it can be seen from Table I that, for the basic architecture, S_X is the reciprocal of S_Y, and F_X is the reciprocal of F_Y, so that S_network=F_network=1. The specificity of pathway X can be increased by changing the magnitude of certain parameters (increasing a₂ or decreasing b₂, for example), but in so doing the specificity of Y decreases correspondingly (see Figure 2B and C).

Two other useful network measurements are mutual specificity (and mutual fidelity), properties that exist if all pathways in the network have specificity (fidelity) greater than 1. The basic architecture never exhibits mutual specificity or mutual fidelity.

Analysis of insulating mechanisms: compartmentalization

Real cellular signaling networks that share components typically contain one or more insulating mechanisms that are thought to contribute to specificity and fidelity (Tan and Kim, 1999; Schwartz and Madhani, 2004). In compartmentalization, different pathways are localized to different cellular compartments, or to different spatial locations within the cell (Figure 2D) (Smith and Scott, 2002; White and Anderson, 2005). The extent of leaking between the two pathways is determined by the efficiency of compartmentalization. For example, assume that the pathway-specific components of pathway X are localized to the nucleus, while those of pathway Y are localized to the cytosol. Although the shared kinase, x₁, is found in both compartments (x₁^N is the nuclear pool and x₁^C is the cytosolic pool), x₁ activated by x₀ in the nucleus is likely to encounter target x₂, which is also in the nucleus; it will only encounter target y₂ if it diffuses into the cytosol before it is deactivated. Thus, crossover between the two pathways happens when kinase x₁ leaks in or out of the nucleus. D_x is the coefficient for the rate at which x₁ exits the nucleus and enters the cytosol, and D_y is the rate constant for x₁ leaving the cytosol and entering the nucleus. D_x and D_y can be considered as pseudo-diffusion rate constants, or exchange rate constants. The parameters d₁^x and d₁^y are the deactivation constants for x₁ in the nucleus and cytosol, respectively.

Again, assuming weak activation, the network can be modeled with linear ODEs and precise solutions for specificity and fidelity obtained (see Table I and Supplementary information). The specificity of this network is

It can be seen that network specificity is greater than in the basic architecture, and is maximized if the exchange rates D_x and D_y are small compared to the deactivation rates d₁^x and d₁^y. Compartmentalization can also provide both mutual specificity and mutual fidelity, as long as the exchange rates balance each other (see Table I). The limiting case where D_x=D_y=0 is equivalent to two noninteracting cascades with complete specificity and fidelity. If the leakage coefficients become very large (D_x, D_y ), we again have a fully connected system with a shared element, equivalent to the basic architecture, and S_network=F_network=1.

Role of scaffold proteins

Scaffold proteins, defined here as proteins that bind to two or more consecutively acting components in a pathway, have been shown to enhance the efficiency of signaling, and have also been proposed to augment specificity by several mechanisms (Whitmarsh and Davis, 1998; Levchenko et al, 2000; Burack et al, 2002; Flatauer et al, 2005). In particular, by binding to multiple components of a given pathway, scaffolds may create the equivalent of 'micro-compartments' (Harris et al, 2001). That is, if the reactions between those components can only happen on the scaffold (or are much more efficient on the scaffold), then it is as if these scaffolded reactions occur in their own compartment, sequestered away from reactions occurring off-scaffold. In this way, scaffolds may prevent their bound components that are shared with other pathways from straying into those pathways, and protect them from intrusions from those pathways.

To model this sequestration mechanism, we use the equations for compartmentalization, with the meaning of some of the terms interpreted differently (see Figure 2E and Table I). First, x₁^N (aNchored x₁) is interpreted to represent kinase x₁ bound to the scaffold and x₁^C (Cytosolic x₁) is unbound x₁, free in solution in the cytosol. The equation for dx₁^N/dt then indicates that the activation of kinase x₁ by signal x₀ occurs on the scaffold and not in solution, while the equation for dx₂/dt indicates that the activation of target x₂ by kinase x₁ also occurs only on the scaffold. In contrast, the corresponding reactions for pathway Y can occur only in solution and not on the scaffold. D_x is the rate constant for the dissociation of x₁ from the scaffold, and D_y is a first-order association constant for the binding of cytosolic x₁ to the scaffolded complex. Leaking between pathways X and Y can occur if x₀-activated x₁ dissociates from the scaffold and encounters y₂, or if x₁ that was activated by y₀ in the cytosol binds to the scaffold (see Figure 2E).

The previous results (see equation (5) and Table I) for specificity and fidelity under compartmentalization also apply to scaffolding: S_X is promoted by a low rate of dissociation of kinase x₁ from the scaffold and a high rate of rebinding; however, these factors reduce S_Y. Obtaining network specificity again requires that deactivation rates be fast relative to exchange rates, so that, for instance, any x₁ that dissociates from the scaffold will be deactivated before it encounters y₂. In this model, d₁^x represents the deactivation of kinase x₁ on the scaffold. Interestingly, one way in which it has been proposed that scaffold proteins might enhance signal transmission is by protecting their bound kinases from the action of phosphatases (Levchenko et al, 2000; Heinrich et al, 2002), equivalent to lowering d₁^x to close to or equal to zero. Although this might indeed enhance the speed, duration and amplitude of X signaling (Heinrich et al, 2002), it would lower both S_Y, F_Y and network specificity.